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Measuring Factor Exposures: Uses and Abuses

Ronen Israel

Principal

Adrienne Ross

Associate

October 2015

A growing number of investors have come to view

their portfolios (especially equity portfolios) as a

collection of exposures to risk factors. The most

prevalent and widely harvested of these risk factors

is the market (equity risk premium); but there are

also others, such as value and momentum (style

premia).

Measuring exposures to these factors can be a challenge. Investors need to understand how

factors are constructed and implemented in their

portfolios. They also need to know how statistical

analysis may be best applied. Without the proper

model, rewards for factor exposures may be

misconstrued as alpha, and investors may be

misinformed about the risks their portfolios truly

face.

This paper should serve as a practical guide for

investors looking to measure portfolio factor

exposures. We discuss some of the pitfalls

associated with regression analysis, and how factor

design can matter a lot more than expected.

Ultimately, investors with a clear understanding of

the risk sources in an existing portfolio, as well as

the risk exposures of other portfolios under

consideration, may have an edge in building better

diversified portfolios.

We would like to thank Cliff Asness, Marco Hanig, Lukasz Pomorski, Lasse Pedersen, Rodney Sullivan, Scott Richardson,

Antti Ilmanen, Tobias Moskowitz, Daniel Villalon, Sarah Jiang and Nick McQuinn for helpful comments and suggestions.

Measuring Factor Exposures: Uses and Abuses 1

Introduction: Why Should Investors Care About Factor Exposures?

Investors have become increasingly focused on

how to harvest returns in an efficient way. A big

part of that process involves understanding the

systematic sources of risk and reward in their

portfolios. “Risk-based investing” generally views

a portfolio as a collection of return-generating

processes or factors. The most straightforward of

these processes is to invest in asset classes, such

as stocks and bonds (asset class premia). Such

risk taking has been rewarded globally over the

long term, and has historically represented the

biggest driver of returns for investors. However,

asset class premia represent just one dimension

of returns. A largely independent, separate source

comes from style premia. Style premia are a set of

systematic sources of returns that are well

researched, geographically pervasive and have

been shown to be persistent. There is a logical,

economic rationale for why they provide a long-

term source of return (and are likely to continue

to do so).1 Finally, they can be applied across

multiple asset classes.2

The common feature of risk-based investing is

the emphasis on improved risk diversification,

which can be achieved by identifying the sources

of returns that are underrepresented in a

portfolio. Investors who understand what risk

sources their portfolios are exposed to (and the

magnitude of these exposures) may be better

suited to evaluate existing and potential

managers. Without an understanding of portfolio

risk factor exposures, how else would investors be

able to tell if their value manager, for example, is

actually providing significant value exposure? Or

1 See “How Can a Strategy Still Work if Everyone Knows About It?” accessed September 23, 2015, www.aqr.com/cliffs-perspective/how-

can-a-strategy-still-work-if-everyone-knows-about-it. 2 Applying styles across multiple asset classes provides greater

diversification. In addition, the effectiveness of styles across asset

classes helps dissuade criticisms of data mining. Asness, Moskowitz and

Pedersen (2013); Asness, Ilmanen, Israel and Moskowitz (2015). Past

performance is not indicative of future results.

whether a manager is truly delivering alpha, and

not some other factor exposure? Or even, whether

a new manager would be additive to their existing

portfolio?

These are important questions for investors to

answer, but quantifying them may be difficult.

There are many ways to measure and interpret

the results of factor analysis. There are also many

variations in portfolio construction and factor

portfolio design. Even a single factor such as

value has variations that an investor should

consider — it can be applied as a tilt to a long-

only equity portfolio,3 or it can be applied in a

“purer” form through long/short strategies; it can

be based on multiple measures of value, or a

single measure such as book-to-price; or it can

span multiple asset classes or geographies.

Simply put, even two factors that aim to capture

the same economic phenomenon can differ

significantly in their construction — and these

differences can matter.

In this paper, we discuss some of the difficulties

associated with measuring and interpreting

factor exposures. We explore the pitfalls of

regression analysis, describe the differences

associated with academic versus practitioner

factors, and outline various choices that can

affect the results. We hope that after reading this

paper investors will be better able to measure

portfolio factor exposures, understand the results

of factor models and, ultimately, determine

whether their portfolios are accessing the sources

of return they want in a diversified manner.

A Brief History of Factors

Asset pricing models generally dictate that risk

factors command a risk premium. Modern

Portfolio Theory quantifies the relationship

between risk and expected return, distinguishing

3 The long-only style tilt portfolio will still have significant market

exposure. This type of style portfolio is often referred to as a “smart beta”

portfolio.

http://www.aqr.com/cliffs-perspective/how-can-a-strategy-still-work-if-everyone-knows-about-ithttp://www.aqr.com/cliffs-perspective/how-can-a-strategy-still-work-if-everyone-knows-about-it

2 Measuring Factor Exposures: Uses and Abuses

between two types of risks: idiosyncratic risk (that

which can be diversified away) and systematic

risk (such as market risk that cannot be

diversified away). The Capital Asset Pricing

Model (CAPM) provides a framework to evaluate

the risk premium of systematic market risk.4 In

the CAPM single-factor world, we can use linear

regression analysis to decompose returns into two

components: alpha and beta. Alpha is the portion

of returns that cannot be explained by exposure

to the market, while beta is the portion of returns

that can be attributed to the market.5 But studies

have shown that single-factor models may not

adequately explain the relationship between risk

and expected return, and that there are other risk

factors at play. For example, under the

framework of Fama and French (1992, 1993) the

returns to a portfolio could be better explained by

not only looking at how the overall equity market

performed but also at the performance of size and

value factors (i.e., the relative performance

between small- and large-cap stocks, and between

cheap and expensive stocks). Adding these two

factors (value and size) to the market created a

multi-factor model for asset pricing. Academics

have continued to explore other risk factors, such

as momentum6 and low-beta or low risk,7 and

have shown that these factors have been effective

in explaining long-run average returns.

In general, style premia have been most widely

studied in equity markets, with some classic

examples being the work of Fama and French

referenced above. For each style, they use single,

simple and fairly standard definitions — they are

described in Exhibit 1.8

4 CAPM says the expected return on any security is proportional to the

risk of that security as measured by its market beta. 5 More generally, the economic definition of alpha relates to returns that cannot be explained by exposure to common risk factors (Berger, Crowell,

Israel and Kabiller, 2012). 6 Jegadeesh and Titman (1993); Asness (1994). 7 Black (1972); Frazzini and Pedersen (2014).

8 Specifically, these factors are constructed as follows: SMB and HML

are formed by first splitting the universe of stocks into two size

categories (S and B) using NYSE market-cap medians and then splitting

Exhibit 1: Common Academic Factor Definitions

HML “High Minus Low”: a long/short measure

of value that goes long stocks with high

book-to-market values and short stocks with low book-to-market values

UMD “Up Minus Down”: a long/short measure

of momentum that goes long stocks with high returns over the past 12 months

(skipping the most recent month) and

short stocks with low returns over the

same period

SMB “Small Minus Big”: a long/short measure

of size that goes long small-market-cap

stocks and short large-market-cap stocks

Assessing Factor Exposures in a Portfolio

Using these well-known academic factors, we can

analyze an illustrative portfolio’s factor

exposures. But before we do, we should

emphasize that the factors studied here are not a

definitive or exhaustive list of factors. We should

also emphasize that different design choices in

factor construction can result in very different

measured portfolio exposures. Indeed, the fact

that you can still get large differences based on

specific design choices is much of our point; we

will revisit these design choices later in the paper.

A common approach to measuring factor

exposures is linear regression analysis; it

describes the relationship between a dependent

variable (portfolio returns) and explanatory

stocks into three groups based on book-to-market equity [highest

30%(H), middle 40%(M), and lowest 30%(L), using NYSE

breakpoints].The intersection of stocks across the six categories are

value-weighed and used to form the portfolios SH(small, high book-to-

market equity (BE/ME)), SM(small, middle BE/ME), SL (small, low BE/ME),

BH(big, high BE/ME), BM(big, middle BE/ME), and BL (big, low BE/ME),

where SMB is the average of the three small stock portfolios (1/3 SH +

1/3 SM + 1/3 SL) minus the average of the three big stock portfolios (1/3 BH + 1/3 BM + 1/3 BL) and HML is the average of the two high book-to-

market portfolios (1/2 SH+ 1/2 BH) minus the average of the two low

book-to-market portfolios (1/2 SL + 1/2 BL). UMD is constructed similarly

to HML, in which two size groups and three momentum groups [highest

30% (U), middle 40% (M), lowest 30% (D)] are used to form six portfolios

and UMD is the average of the small and big winners minus the average of

the small and big losers.


variables (risk factors).9 It can be done with one

risk factor or as many as the portfolio aims to

capture. If the portfolio captures multiple styles,

then multiple factors should be used. If the

portfolio is a global multi–asset style portfolio,

then the relevant factors should cover multiple

assets in a global context. Ideally, the factors used

should be similar to those implemented in the

portfolio, or at least one should account for those

differences in assessing the results. For example,

long-only portfolios are more constrained in

harvesting style premia as underweights are

capped at their respective benchmark weights. In

contrast, long/short factors (and portfolios) are

purer in that they are unconstrained. These

differences should be accounted for when

performing and interpreting factor analysis.

For this analysis, we examine a hypothetical

long-only equity portfolio that aims to capture

returns from value, momentum and size.

Specifically, the portfolio is constructed with

50/50 weight on simple measures of value (book-

to-price, using current prices10) and momentum

(price return over the last 12 months) within the

small-cap universe.11 In practice an investor may

not know the portfolio exposures in advance, but

since our goal is to illustrate how to best apply the

analysis, we will proceed as if we do.

9 Note that regressions are in essence just averages over a given period

and will not provide any insight into whether a manager varies factor exposures over time. To understand how factor exposures vary over time

you can look at rolling betas, ideally using at least 36 months of data. But

the tradeoff is that some, perhaps a lot, of this variation may in fact be

random noise. Past performance is not indicative of future results. 10 Fama-French HML uses lagged prices. See section on “other factor

design choices.” 11

See Frazzini, Israel, Moskowitz and Novy-Marx (2013), for more detail

on how to construct a multi-style portfolio. Note that we have followed a

similar multi-style portfolio construction approach here. To build our

portfolio, we rank stocks based on simple measures for value (book-to-

price using current prices) and momentum (price return over the last 12

months) within the U.S. small-cap universe (Russell 2000). We compute a composite rank by applying a 50% weight to value and 50% to

momentum. We then select the top 25% of stocks with the highest

combined ranking and weight the stocks in the resulting portfolio via a

50/50 combination of each stock’s market capitalization and

standardized combined rank. Portfolio returns are gross of transaction

costs, un-optimized and undiscounted. The portfolios are rebalanced

monthly.

We start with a simple one-factor model and then

add the additional factors that the portfolio aims

to capture. We analyze style exposures using

academic factors (over practitioner factors) for

simplicity and illustrative purposes. The

performance characteristics of the portfolio and

factors used are shown in Exhibit 2, which shows

that the portfolio returned an annual 13.5% in

excess of cash on average from 1980–2014.

We can use these returns and betas from

regression analysis to decompose portfolio excess

of cash returns (𝑅𝑖 − 𝑅𝑓). 12 The first regression

model we look at is the CAPM with the market as

the only factor.13

(1) (𝑅𝑖 − 𝑅𝑓) = 𝛼 + 𝛽𝑀𝐾𝑇(𝑅𝑀𝐾𝑇 − 𝑅𝑓) + 𝜀

Or roughly,

Portfolio Returns in Excess of Cash =

Alpha + Beta x Market Risk Premium14

The results in Exhibit 3 show that the portfolio

had a market beta of 0.96 (based on Model 1 in

12

One of the most common mistakes in running factor analysis is to

forget to take out cash from the returns of the left- and right-hand side

variables. For a long-only factor or portfolio, such as the market, one must

explicitly do that. A long/short factor is a self-financed portfolio whose

returns are already in excess of cash.

13 We have also included an error term ( ), which is the difference between

actual realized returns and expected returns. More specifically, the error

term captures the unexplained component of the relationship between the

dependent variable (e.g., the portfolio excess returns) and explanatory

variables (e.g., the market risk premium). 14

All risk premia in this paper are returns in excess of cash.

Exhibit 2: Hypothetical Performance Statistics

January 1980–December 2014

Note: All returns are arithmetic averages. Returns are in excess of cash. Source: AQR, Ken French Data Library. The portfolio is a hypothetical simple 50/50 value and momentum long-only small-cap equity portfolio, gross of fees and transaction costs, and excess of cash. The portfolio is rebalanced monthly. The academic explanatory variables are the contemporaneous monthly Fama-French factors for the market (MKT-RF), value (HML), momentum (UMD) and size (SMB). The market is the value-weight return of all CRSP firms. Hypothetical data has inherent limitations some of which are discussed herein.


Part A). This means — not surprisingly, as the

portfolio is long-only — that the portfolio had

meaningful exposure to the market. We also

know (from Exhibit 2) that over this period the

equity market has done well, delivering 7.8%

excess of cash returns. As a result, we can see (in

Part B of Exhibit 3) that the portfolio’s positive

exposure to the market contributed 7.4% to

overall returns,15 and that 6.1% was “alpha” in

excess of market exposure.

The same framework can be applied for multiple

risk factors. Our first multivariate regression adds

the value factor.

(2) (𝑅𝑖 − 𝑅𝑓) =

𝛼 + 𝛽𝑀𝐾𝑇(𝑅𝑀𝐾𝑇 − 𝑅𝑓) + 𝛽𝐻𝑀𝐿(𝑅𝐻𝑀𝐿) + 𝜀

The results under Model 2 show that the portfolio

had positive exposure to value (with a beta of

0.43), which means that the portfolio on average

bought cheap stocks.16 Because value is a

historically rewarded long-run source of returns,

having positive exposure benefited the portfolio,

with value contributing 1.6% to portfolio returns

(𝐻𝑀𝐿 𝑏𝑒𝑡𝑎 × 𝐻𝑀𝐿 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 0.43 × 3.6%).

Next we add the momentum factor in Model 3.

(3) (𝑅𝑖 − 𝑅𝑓) =

𝛼 + 𝛽𝑀𝐾𝑇(𝑅𝑀𝐾𝑇 − 𝑅𝑓)+ 𝛽𝐻𝑀𝐿(𝑅𝐻𝑀𝐿) +

+ 𝛽𝑈𝑀𝐷(𝑅𝑈𝑀𝐷) + 𝜀

As one would expect, we see that the momentum

loading is positive (with a beta of 0.09), which

means that the portfolio on average bought

recent winners. But the magnitude of this

exposure is smaller than expected for a portfolio

that aims to capture returns from momentum

investing. It seems that momentum only

contributed 0.6% to portfolio returns (𝑈𝑀𝐷 𝑏𝑒𝑡𝑎 ×

𝑈𝑀𝐷 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 0.09 × 7.3%), while value

15 Market beta × market risk premium = 0.96 × 7.8%. 16 Even though value has a negative univariate correlation with the

portfolio (as seen in Exhibit 2), we can see that after controlling for

market exposure (in Exhibit 3), the portfolio loads positively on value. We

will discuss the importance of multivariate over univariate regressions for

a multi-factor portfolio later in the paper.

contributed 1.7%. This may seem odd for a

portfolio that is built with a 50/50 combination of

value and momentum. But we should keep in

mind that we’re still looking at an incomplete

model — one without all the risk factors in the

portfolio. Let’s see what happens when we add

the size variable in our next model (Model 4 in

Exhibit 3).

(4) (𝑅𝑖 − 𝑅𝑓) =

𝛼 + 𝛽𝑀𝐾𝑇(𝑅𝑀𝐾𝑇 − 𝑅𝑓)+ 𝛽𝐻𝑀𝐿(𝑅𝐻𝑀𝐿) +

+ 𝛽𝑈𝑀𝐷(𝑅𝑈𝑀𝐷) + 𝛽𝑆𝑀𝐵(𝑅𝑆𝑀𝐵) + 𝜀

In our final model (which includes all the sources

of return that the portfolio aims to capture), we

still see a small beta on momentum, with the

factor contributing 0.5% to portfolio returns over

the period (𝑈𝑀𝐷 𝑏𝑒𝑡𝑎 × 𝑈𝑀𝐷 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 0.07 ×

7.3%). However, this unintuitive result can be

largely explained by factor design differences.

Stay tuned and we will come back to this issue

later in the paper.17

The good news is that when it comes to the other

factors in Model 4, the results are consistent with

intuition. For size, we see a large positive

exposure (beta of 0.74), which means the portfolio

had meaningful exposure to small-cap stocks.

This exposure contributed 1.2% to portfolio

returns over the period. We also see that after

controlling for size, value had an even larger beta,

which means that it contributed 2.4% to portfolio

returns.

17 See the section on “other factor design choices” where we discuss how HML can be viewed as an incidental bet on UMD; this affects regression

results by lowering the loading on UMD (as HML is eating up some of the

UMD loading that would otherwise exist). We correct for this in Appendix

A, and show a higher loading on UMD. Also see Frazzini, Israel,

Moskowitz and Novy-Marx (2013) and Asness, Frazzini, Israel and

Moskowitz (2014) for more information on the most efficient way to gain

exposure to momentum.


Exhibit 3: Decomposing Hypothetical Portfolio Returns by Factors


Part A: Regression Results

Part B: Portfolio Return Decomposition

Note: All returns are arithmetic averages. The bar chart in Part B uses the factor returns (from Exhibit 2) and factor betas (from Part A) to decompose

portfolio returns. Numbers may not exactly tie out due to rounding.

Source: AQR, Ken French Data Library. AQR analysis based on a hypothetical simple 50/50 value and momentum long-only small-cap equity portfolio,

gross of fees and transaction costs, and excess of cash. The portfolio is rebalanced monthly. The academic explanatory variables are the

contemporaneous monthly Fama-French factors for the market (MKT-RF), value (HML), momentum (UMD) and size (SMB). The market is the value-weight return of all CRSP firms. Hypothetical data has inherent limitations some of which are discussed herein.

Model 1 (Market Control)

Model 2 (Add HML)

Model 3 (Add UMD)

Model 4 (Add SMB)

Alpha (ann.) 6.1% 3.8% 2.9% 1.8%

t-statistic 3.6 2.5 1.9 2.2

Market Beta 0.96 1.05 1.07 0.99

t-statistic 31.1 35.7 36.0 61.5

HML Beta 0.43 0.46 0.65

t-statistic 9.8 10.3 26.4

UMD Beta 0.09 0.07

t-statistic 3.0 4.6

SMB Beta 0.74

t-statistic 32.2

R2 0.70 0.75 0.76 0.93

7.4%8.2% 8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(Add HML)

Model 3

(Add UMD)

Model 4

(Add SMB)

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

An

nu

al E

xc

es

s R

etu

rn

7.5% 7.9% 7.9% 7.4%

0.9% 0.8% 1.3%

-0.2% -0.3%

0.8%

0.0%

-1.3%

-1.0% -1.8%

-3.0%

-1.5%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(Add HML)

Model 3

(Add UMD)

Model 4

(Add SMB)

Market

HML

UMD

SMB

Alpha


Ultimately, in interpreting the results of

regression analysis investors should focus on the

model that includes the systematic sources of

returns that the portfolio aims to capture; in this

case, it would be Model 4. For portfolios that

capture styles in an integrated way, it’s important

to include multiple factors to control for the

correlation between factors. In other words, to

take into account how factors are related to each

other. It is well known that value and momentum

are negatively correlated, and portfolios formed

in an integrated way can take advantage of this.

Focusing on how value performs stand-alone (i.e.,

Model 2) has little implication on how value adds

to a portfolio that combines value with other

factors synergistically (i.e., Model 4). One of the

benefits of multi-factor investing is the relatively

low correlations factors have with each other,

making the “whole” more efficient than the sum

of its parts.

Alpha vs. Beta

While betas are important in understanding

factor exposures in a portfolio, alpha can be

useful in analyzing manager “skill.” It’s important

that investors are able to tell whether a manager

is actually providing alpha, above and beyond

their intended factor exposures. But this means

that they need to be sure that they’re using the

correct model when analyzing factor exposures.

Without the proper model, rewards for factor

exposures may be misconstrued as alpha. This

can lead to suboptimal investment choices, such

as hiring a manager that seems to deliver “alpha,”

but really just provides simple factor tilts.

To illustrate this point we can look at the alpha

estimates in Exhibit 3.18 By looking at each model

on a step-wise basis, we can see how the inclusion

of additional risk factors reduces alpha

significantly; in other words, alpha has been

18

It’s important to caveat that even with a large number of observations

(i.e., more than five years), alpha can be difficult to assess with conviction.

replaced by factor exposures. When the market is

the only factor (Model 1) it seems as though the

portfolio has significant alpha at 6.1%, but when

we add the other risk factors we see that alpha is

reduced to 2.9% with value and momentum, and

finally to 1.8% with all four factors.19 These

results have important implications — if you

don’t control for multiple exposures in a multi-

factor portfolio, then excess returns will look as if

they are mostly alpha.

But it’s also important to note that “alpha”

depends on what is already in your portfolio. For

any portfolio, adding positive expected return

strategies that are uncorrelated to existing risk

exposures can provide significant portfolio alpha.

For the market portfolio, adding value and

momentum exposures can have the same effect

as adding alpha (as both represent new, more

efficient sources of portfolio returns).20 Along the

same lines, adding momentum to a value

portfolio can provide significant alpha.

The main point is this: in order to determine

whether such a factor can be “alpha to you,” an

investor must first determine which factors are

already present in their existing portfolio — those

that are not can potentially be alpha.

Understanding Factor Exposures: A Deeper Dive

We now turn to a more detailed discussion of the

statistics involved in regression analysis. We

hope these details will help investors better

understand and interpret the results of regression

models.

The Mechanics of Beta

Investors looking to analyze portfolio exposures

often look at betas of regression results. Beta

19 Note that alpha goes away when you include a “purer” measure of value

based on current price; this is shown in Appendix A and described in the

section “other factor design choices.” 20

See Berger, Crowell, Israel and Kabiller (2012) in which they discuss

the concept of “alpha to you.”


measures the sensitivity of the portfolio to a

certain factor. In the case of market beta, it tells

us how much a security or portfolio’s price tends

to change when the market moves. From a

mathematical standpoint, the beta for portfolio i

is equal to its correlation with the market times

the ratio of the portfolio’s volatility to the

market’s volatility.21

or,

𝑓𝑎𝑐𝑡𝑜𝑟 𝑏𝑒𝑡𝑎 = 𝑓𝑎𝑐𝑡𝑜𝑟 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 ×

(𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦

𝑓𝑎𝑐𝑡𝑜𝑟 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦)

Since volatility varies considerably across

portfolios, comparisons of betas can be

misleading. For the same level of correlation, the

higher a portfolio’s volatility, the higher its beta.

Let’s see why this matters. Suppose an investor is

comparing value exposure for two different

portfolios: portfolio A is a defensive equity

portfolio (with lower volatility) and portfolio B is

a levered equity portfolio (with higher volatility).

It could be the case that portfolio B has a higher

value beta, which would seem to indicate that it

has higher value exposure. However, the higher

beta could be a result of portfolio B’s higher

volatility, rather than more meaningful value

exposure (assuming the same level of correlation

between both portfolios and the value factor).

When investors fail to account for different levels

of volatilities between portfolios, they may

conclude that one portfolio is providing more

value exposure than another, which it does in

notional terms — but in terms of exposure per

unit of risk, that may not be the case.

21

This equation applies for betas using a univariate regression, i.e., with a

single right-hand side variable. Multivariate regression betas may differ

from univariate betas because they control for the other right-hand side

variables, which means that they take correlations into account.

This approach can also be extended to

comparisons of different factors for the same

portfolio. Looking back at Exhibit 3 under Model

4, we can compare the loadings on value and

momentum. One would expect similar betas on

these factors as the portfolio is built to target each

equally (with 50/50 weight).22 But even with

similar correlation with the portfolio, value has a

meaningfully higher loading (looking at Model 4).

Does this mean that value contributes more than

momentum? Not necessarily as we need to

account for their differing levels of volatility. For

the same level of correlation, the higher a factor’s

volatility, the lower its beta. Put differently, the

lower beta on UMD versus HML is partly driven

by differing volatility levels23 — from Exhibit 2 we

see that UMD had volatility of 15.8%, while HML

had volatility of 10.5%.

But investors can make adjustments to allow for

more direct beta comparisons. When comparing

factors for the same portfolio, the impact of

differing volatilities should be eliminated; this

can be done by volatility scaling the right-hand

side (RHS) factors such that they all realize the

same volatility. And for those looking to compare

betas across portfolios (on a risk-adjusted basis),

they can look at correlations, which are invariant

to volatility and can be compared more directly

across portfolios with different volatilities.24

22 Some investors may be familiar with the work that Sharpe did on style

analysis (1988, 1992). This approach constrains the regression so that

the coefficients are positive and sum to one. In this case, the coefficients

(or betas) can be used as weights in building the ‘replicating’ portfolio. In

other words, a portfolio with factor weights equal to the weighted

average of the coefficients should behave similar in terms of its returns. 23

The lower relative loading is partly driven by differing volatilities, but it

is also a result of the fact that HML can be viewed as an incidental bet on

both value and momentum. We correct for this by using a “purer” measure of value; this is shown in Appendix A and described in the section “other

factor design choices.” 24 Though for a multi-factor portfolio, investors should focus on partial

correlations, which provide insight into the relationship between two

variables while controlling for a third. Alternatively, for a long-only

portfolio investors can look at correlations using active returns; that is,

net out the market or benchmark exposure.


Portfolio Risk Decomposition

Betas from regression analysis can also be used in

portfolio risk attribution. This approach is best

thought of as variance decomposition, and is

done by using factor beta, factor volatility,

portfolio volatility and factor correlations.25 For

example, from Exhibit 2 and 3 we see that the

market factor had average volatility of 15.6% and

a market beta of 0.96 (based on Model 1). This

tells us that the market factor dominates the risk

profile of the portfolio, contributing an estimated

14.9% risk to the portfolio

(√market beta2 × market volatility2 =

√0.962 × 15.6%2).26 Given that overall portfolio risk

is 17.8%, we can estimate the proportion of

variance that is being driven by market exposure

(market variance contribution

portfolio variance) = (

14.9%2

17.8%2) = 0.70. This means

that roughly 70% of portfolio variance can be

attributed to the market risk factor.27 But there is

an interesting application of this result: 0.70 is

the same as the R2 measure for Model 1 (shown in

the final row of the regression table in Exhibit 3).

We will now discuss R2 in more detail.

The R2

Measure: Model Explanatory Power

The R2 measure provides information on the

overall explanatory power of the regression

model. It tells us how much of returns are

explained by factors included on the right-hand

side of the equation. Generally, the higher the R2

the better the model explains portfolio returns.

We can see from the R2 measure at the bottom of

the table in Exhibit 3 that multivariate analysis is

more effective (than univariate) at explaining

25

This approach is similar to decomposing portfolio risk by using portfolio

weights, correlation and volatility estimates. We have included an

example of how to do this for a simple two factor portfolio in Appendix B. 26 Note that volatility is the square root of variance. 27 In this case the idiosyncratic, asset-specific risk would account for

30% of the overall variance of the portfolio. This example focuses on a

single-factor model where we can ignore factor correlations. If we were to

apply the same approach for a multi-factor model, factor correlations

would matter and we would need to incorporate the covariance matrix.

This approach requires matrix algebra and is computationally intensive, so

we have omitted the calculation.

returns for a multi-factor portfolio. In particular,

we see in the final column of the table that the

inclusion of additional risk factors has improved

the explanatory power of the model (that is, how

much of portfolio variance is being captured by

these factors), with the R2 improving from 0.70 to

0.93.28

The t-statistic: A Measure of Statistical

Significance

While beta tells us whether a factor exposure is

economically meaningful (and how much a factor

may contribute to risk and returns), it doesn’t tell

us whether the factor exposure is statistically

significant. Just because a portfolio has a high

beta coefficient to a factor doesn’t mean it’s

statistically different than a portfolio with a zero

beta, or no factor exposure. As such, it’s

important to look at the t-statistic. This measure

tells us whether a particular factor exposure is

statistically significant. It is a measure of how

confident we are about our beta estimates.29

When the t-statistic is greater than two, we can

say with 95% confidence (or a 5% chance we are

wrong) that the beta estimate is statistically

different than zero.30 In other words, we can say

that a portfolio has significant exposure to a

factor.

Looking back at the momentum factor, even

though the portfolio may not have an

economically meaningful beta (at 0.07 in Model

4), we can see that it is statistically significant

28 Note that it’s more accurate to look at the adjusted R2 when comparing

models with a different number of explanatory variables. By construction,

the R2 will never be lower and could possibly be higher when additional

explanatory variables are included in the regression; and the adjusted R2

corrects for that. When there are a large number of observations the two

measures will be similar; this is the case with our regression as we use

monthly data over 35 years (meaning a large sample size with 420

observations). 29 It’s important to note that the t-statistic increases with more observations; that is, as the sample size grows very large we are more

certain about our beta estimates. 30 A t-statistic of two generally represents a reasonable standard of

significance (implies statistical significance at a 95% confidence interval

under the assumption of a normal distribution) if no look-ahead bias.

Generally, the higher the t-statistic the more confident we can be about

our beta estimates.


(with a t-statistic greater than two). The t-statistic

is an especially important measure for comparing

portfolios with different volatilities.

At the end of the day, both beta and t-statistics

provide valuable information when assessing

factor exposures. A factor exposure that is both

economically meaningful and statistically

significant (reliable) means you can count on it

impacting your portfolio in a big way. An

exposure that is only economically meaningful

but not reliable could impact you in a big way, but

with a high degree of uncertainty. Finally, an

exposure that is small but reliable means you can

expect (with greater certainty) that it will impact

your portfolio, but only in a small way. While an

investor may not care a lot about this last

application, it’s still worth understanding when

analyzing the regression output.

Factor Differences: Academics vs. Practitioners

So far we have focused on factor analysis and

how to interpret the results. But the results of

factor analysis are highly influenced by how

factors are formed. There are many differences

between the ways factors are measured from an

academic standpoint versus how they get

implemented in portfolios. Investors should be

aware that not all factors are the same, even those

attempting to measure essentially the same

economic phenomenon — and these differences

can matter. We focus here on design decisions

that can have a meaningful impact on the results

of factor analysis.

Implementability

At a basic level, academic factors do not account

for implementation costs. They are gross of fees,

transaction costs and taxes. They do not face any

of the real-world frictions that implementable

portfolios do. Essentially, they are not a perfect

representation of how factors get implemented in

practice.

Differences in implementation approaches may

be reflected in factor model results. Even if a

portfolio does a perfect job of capturing the

factors, it could still have negative alpha in the

regression, which would represent

implementation differences associated with

capturing the factors. For example, if you

compare a portfolio that faces trading costs

versus one that doesn't, clearly the former will

underperform the latter, possibly implying

negative alpha. In fact, this is exactly what we see

when we look at a composite of mutual funds —

these results are shown in Appendix C. When

looking at a live portfolio against academic

factors, investors should not be surprised by

negative alpha. In these cases, investors should

either use practitioner factors on the RHS, or just

focus on beta comparisons because trading costs

and other implementation issues do not affect

these estimates.31

Investment Universe

Academic factors (such as those used here) span a

wide market capitalization range and are, in fact,

overly reliant on small-cap stocks or even micro-

cap stocks (we will explain this in greater detail in

the next section). The factors include the entire

CRSP universe of approximately 5,000 stocks.

Many practitioners would agree that a trading

strategy that dips far below the Russell 3000 is

not a very implementable one, and this is likely

where most of the bottom two quintiles in the

academic factors fall.

Practitioners mainly focus on large- to mid-cap

universes for investability reasons. For portfolios

that provide exposure to the large-cap universe,

academic factors may not be an accurate

31

Specifically, these implementation issues drop out of the covariance.

Implementation issues, such as fees and transaction costs, are relatively

stable components (constants), which mathematically don’t matter much

for higher moments such as covariance.


representation of desired exposures. Given that

academic factors span a wide range of market

capitalization, factor analysis results will be

highly impacted by the influence of some other

part of the capitalization range — a range that is

not being captured in the portfolio by design.

Factor Weighting

Generally, academic factors are formed using an

intersection of size and their particular factor

(value, in the case of HML).32 For the factors

described in Exhibit 1, a stock’s size is determined

by the median market capitalization, which

means a roughly equal number of stocks are

considered “big” and “small.”33 The factors are

formed by giving equal capital weight to each

universe, which given the higher risk of small

stocks likely means that an even greater risk

weight and contribution comes from small stocks.

Practitioners generally take views on the entire

universe, assigning larger positive weights to the

stocks that rank more favorably on some

measure, and larger negative weights to the less

favorable stocks.

Industries

Academic factors do a simple ranking across

stocks, and in doing so implicitly take style bets

within and across industries (also across

countries in international portfolios), without any

explicit risk controls on the relative contributions

of each. In contrast, the factors implemented by

practitioners may differentiate stocks within and

across industries (i.e., industry views). They are

designed to capture and target risk to both

independently. This distinction can result in a

more diversified portfolio, one without

unintended industry concentrations.

32

See footnote 8 for more information on how the academic factors are

constructed. 33

Despite its large number of stocks, the small-cap group contains

roughly 10% of the market-cap of all stocks (Fama and French, 1993).

Risk Targeting

Risk targeting is a technique that many

practitioners use when constructing factors; this

approach dynamically targets risk to provide

more consistent realized volatility in changing

market conditions. Practitioners also build

market (or beta) neutral long/short portfolios.

Academic factors typically do not utilize risk

targeting as their factors are returns to a $1

long/$1 short portfolio, whose risk and market

exposures can vary. The effect of this can be seen

in Exhibit 4, which shows how HML has

significant variation in market exposure over

time.34

Exhibit 4: Varying Market Exposure of HML Over

Time

-1

-0.5

0

0.5

1

Ro

llin

g 3

6m

on

th B

eta

HML Market Beta1926-2014

Source: AQR, Ken French Data Library. Analysis based on the market

(MKT-RF) and HML portfolios. The market is the value-weight return of all

CRSP firms.

Multiple Measures of Styles

While stocks selected using the traditional

academic value measure perform well in

empirical studies, there is no theory that says

book-to-price is the best measure for value. Other

measures can be used and applied

simultaneously to form a more robust and reliable

view of a stock’s value. For example, investors

can look at a variety of other reasonable

34

Note that this graph is meant to be descriptive of the types of issues

that may arise when analyzing non risk-targeted portfolios. We cannot

say for certain how much of the relation shown here is noise, or if it is

predictable.


fundamentals, including earnings, cash flows,

and sales, all normalized by some form of price.

Factors that draw on multiple measures of value

can significantly improve performance, as shown

in Exhibit 5.35

The same intuition applies for other styles. For

example, momentum factors that include both

earnings momentum and price momentum may

be more robust portfolios.

Other Factor Design Choices

Other design decisions can have a meaningful

impact on returns. Looking at the case of value,

Fama–French construct HML using a lagged

value for price that creates a noisy combination

of value and momentum. When forming their

value portfolio on book-to-price, they use the

price that existed contemporaneously with the

book value, which due to financial reporting can

be lagged by 6 to 18 months. So a company that

looked expensive based on its book value and

price from six months ago and whose stock has

fallen over the past six months should look better

from a valuation perspective (since the price is

lower, and holding book value constant36). Yet, in

35

Asness, Frazzini, Israel and Moskowitz (2014); Asness, Frazzini, Israel

and Moskowitz (2015); Israel and Moskowitz (2013). 36

This is a reasonable assumption. See Asness and Frazzini (2013).

a traditional definition (using lagged prices) the

stock is viewed the same way irrespective of the

price move.

An alternative way of looking at it is to define

value with the current price, which means the

stock is now cheaper. On the other hand if you

incorporate momentum into the process the stock

doesn’t look any better since its price has fallen

over the past six months. Putting the two

together, the stock looks more attractive from a

value perspective, but less attractive from a

momentum perspective, with the net effect

ending up potentially in the same place as the

traditional definition of value. As a result, the

traditional definition can be viewed as an

incidental bet on both value and momentum; in

fact, empirically the traditional definition of

value ends up being approximately 80% pure

value (current price) and 20% momentum.37

In order to correct for this noisy combination of

value and momentum, Asness and Frazzini (2013)

suggest replacing the 6- to 18-month lagged price

with the current price to compute valuation ratios

that use more updated information. Measuring

HML using current price (what they call “HML

Devil”) eliminates any incidental exposure to

37

Asness and Frazzini (2013).

Exhibit 5: Design Decisions Are Important in Portfolio Construction

Source: Frazzini, Israel, Moskowitz and Novy-Marx (2013). Book-to-price is defined using current price. The multiple measures of value include book-to-

price, earnings-to-price, forecasted earnings-to-price, cash flow-to-enterprise value, and sales-to-enterprise value.

Hypothetical Average Excess of Russell 1000 Annual ReturnsJanuary 1980 – December 2012

0%

1%

2%

Simple B/P Value Portfolio Using Multiple Measures of Value


momentum, resulting in a better proxy for true

value, while still using information available at

the time of investing.

This factor design choice is especially important

when interpreting regression results. When

regressing a portfolio of value and momentum on

UMD and HML (using the traditional academic

definition), it will appear that UMD has a lower

loading, as HML is eating up some of the UMD

loading that would otherwise exist. This is exactly

what we saw in Exhibit 3, where UMD had a very

low loading. However, if HML is defined using

current price (as is the case with HML Devil), the

value loading will no longer have exposure to

momentum and any momentum exposure in the

portfolio will go directly into UMD, thus raising

its loading. This is consistent with what we see

when we make the HML Devil correction to the

analysis from Exhibit 3: the UMD loading

increases from 0.07 to 0.32; these results are

shown in the Appendix in Exhibit A1.

In this section we have discussed a few factor

differences that can meaningfully affect the

results of factor analysis. As a result, we

encourage investors to be aware of these

differences when interpreting regression results.

Concluding Remarks

Market exposure has historically rewarded long-

term investors, but market risk is only one

exposure among several that can deliver robust

long-term returns. Measuring exposure to risk

factors can be a challenge: factors can be formed

multiple ways and statistical analysis is ridden

with nuances. Ultimately investors who

understand how to measure factor exposures may

be better able to build truly diversified portfolios.

The following summary points are useful for

investors to think about when decomposing

portfolios into risk factors:

Even a single factor such as value has

variations that an investor should consider:

there are many differences between how

factors are constructed from an academic

standpoint versus how they are implemented

in portfolios. In conducting factor analysis,

investors should ask themselves: What

exactly are these factors I’m using? Are they

the same as those I’m getting in my

portfolio? The answers to these questions

affect beta and alpha estimates. Factor

loadings are highly influenced by the design

and universe of factors; and alpha estimates

reflect implementation differences

associated with capturing the factors. For

example, if you compare a portfolio that

faces trading costs versus one that doesn't, it

is not surprising the former will

underperform the latter, and possibly show

negative alpha. When investors want to

compare alphas and betas across managers

they should be sure they are using the factors

being captured in the portfolios. Ultimately,

not accounting for factor exposures properly

can lead to suboptimal investment choices,

such as hiring an expensive manager that

seems to deliver “alpha,” but really just

provides simple factor tilts.

There are many things to consider when

performing statistical analysis on portfolios.

For portfolios with more than one risk factor,

multivariate models are most appropriate.

Investors should consider t-statistics, not just

betas, to assess factor exposures, especially

when comparing portfolios with different

volatilities.

In order to determine whether a certain

factor exposure can be “alpha to you,” an

investor must first determine which factors

are already present in their existing portfolio

— those that are not can potentially be alpha.


Appendix A | Correcting for HML Devil

Exhibit A1: Decomposing Hypothetical Portfolio Returns by Factors



Part B: Return Decomposition

Source: AQR analysis based on a hypothetical simple 50/50 value and momentum long-only small-cap equity portfolio, gross of fees and transaction costs, and excess of cash. The portfolio is rebalanced monthly. The academic explanatory variables are the contemporaneous monthly academic factors

for the market (MKT-RF), value (HML Devil), momentum (UMD), and size (SMB). The portfolio returned 13.5% in excess of cash on average over the

period, the market returned 7.8% excess of cash, HML Devil returned 3.3%, UMD returned 7.3% and SMB returned 1.6%. The market is the value-

weight return of all CRSP firms. Hypothetical data has inherent limitations some of which are discussed herein.


Model 2 (Add HML Devil)

Model 3 (Add UMD)

Model 4 (Add SMB)

Alpha (ann.) 6.1% 5.2% 1.7% 0.7%

t-statistic 3.6 3.2 1.1 0.7

Market Beta 0.96 0.98 1.04 0.94

t-statistic 31.1 32.8 35.5 50.0

HML Devil Beta 0.22 0.48 0.61


UMD Beta 0.29 0.32

t-statistic 7.3 12.9

SMB Beta 0.68

t-statistic 25.1

R2 0.70 0.72 0.75 0.90

7.4% 7.6% 8.1% 7.3%

0.7%1.6%

2.0%

2.1% 2.4%

1.1%

6.1%5.2%

1.7%0.7%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

An

nu

al E

xc

es

s R

etu

rn

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

HML Devil

(add HML Devil)

7.5% 7.9% 7.9% 7.4%

0.9% 0.8% 1.3%

-0.2% -0.3%

0.8%

0.0%

-1.3%

-1.0% -1.8%

-3.0%

-1.5%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(Add HML)

Model 3

(Add UMD)

Model 4

(Add SMB)

Market

HML

UMD

SMB

Alpha


Appendix B | Alternate Method of Hypothetical Portfolio Risk Decomposition

For this example, we use a simple 50/50 value/momentum long/short portfolio.

Step 1: Determine the covariance matrix

Using assumptions on volatility and correlation38

(inputs in blue), we create the covariance matrix.

𝐂𝐨𝐯𝐚𝐫𝐢𝐚𝐧𝐜𝐞(𝐇𝐌𝐋, 𝐔𝐌𝐃) = 𝐂𝐨𝐫𝐫𝐞𝐥𝐚𝐭𝐢𝐨𝐧(𝐇𝐌𝐋, 𝐔𝐌𝐃) × 𝐕𝐨𝐥𝐚𝐭𝐢𝐥𝐢𝐭𝐲(𝐇𝐌𝐋) × 𝐕𝐨𝐥𝐚𝐭𝐢𝐥𝐢𝐭𝐲 (𝐔𝐌𝐃)

= −0.2 × 11% × 16%

= −0.003

Step 2: Determine the variance contribution of each factor

Using capital weights and the covariance matrix from step 1 (shown by the inputs in blue below), we can determine the variance contribution (VAR Contrib.) of each factor.

𝐕𝐀𝐑 𝐂𝐨𝐧𝐭𝐫𝐢𝐛. (𝐇𝐌𝐋) = 𝐖𝐞𝐢𝐠𝐡𝐭(𝐇𝐌𝐋)𝟐 × 𝐕𝐨𝐥𝐚𝐭𝐢𝐥𝐢𝐭𝐲(𝐇𝐌𝐋)𝟐 + 𝐖𝐞𝐢𝐠𝐡𝐭(𝐇𝐌𝐋) × 𝐖𝐞𝐢𝐠𝐡𝐭 (𝐔𝐌𝐃) × 𝐂𝐨𝐯𝐚𝐫𝐢𝐚𝐧𝐜𝐞(𝐇𝐌𝐋, 𝐔𝐌𝐃)

= 50%2 × 11%2 + 50% × 50% × −0.003

= 0.23%

Note: unlike volatility, portfolio variance is additive:

𝐕𝐀𝐑(𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨) = 𝐕𝐀𝐑 𝐂𝐨𝐧𝐭𝐫𝐢𝐛. (𝐇𝐌𝐋) + 𝐕𝐀𝐑 𝐂𝐨𝐧𝐭𝐫𝐢𝐛. (𝐔𝐌𝐃)

= 0.23% + 0.57%

= 0.80%

38

Note that we have used assumptions that are broadly representative of the historical volatilities and correlations for HML and UMD. But the example

applies for any set of assumptions. It is for illustrative purposes only.

Portfolio Inputs

Volatility

Value (HML) 11%

Momentum (UMD) 16%

Correlation Matrix

Value (HML) Momentum (UMD)

Value (HML) 1.0 -0.2

Momentum (UMD) -0.2 1.0

Value (HML)

Momentum (UMD)

Value (HML) 0.012 -0.003


Covariance Matrix

Portfolio Inputs

Volatility Capital Weights

Value (HML) 11% 50%

Momentum (UMD) 16% 50% Variance

Value (HML) 0.23%

Momentum (UMD) 0.57%

Portfolio 0.80%Covariance Matrix

Value (HML) Momentum (UMD)

Value (HML) 0.012 -0.003



Step 3: Determine the percent risk/variance contribution of each factor Finally, using the variance from step 2 we can determine the percent of portfolio variance coming from each

factor.

% 𝐂𝐨𝐧𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧 𝐭𝐨 𝐕𝐚𝐫𝐢𝐚𝐧𝐜𝐞 (𝐇𝐌𝐋) =𝐕𝐀𝐑 𝐂𝐨𝐧𝐭𝐫𝐢𝐛. (𝐇𝐌𝐋)

𝑽𝑨𝑹 (𝑷𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐)

=0.23%

0.80%

≈ 30%

Volatility Capital Weights Variance

Value (HML) 11.0% 50% 0.23%

Momentum (UMD) 16.0% 50% 0.57%

Portfolio 8.9% 100% 0.80%

% Contribution to Variance

Value (HML) 30%

Momentum (UMD) 70%

Portfolio 100%


Appendix C | Applications for a Live Portfolio

In this paper we have focused on a hypothetical portfolio that aims to capture returns from value and

momentum. We have done this for simplicity and illustrative purposes, but the same framework can be

applied for any portfolio. So, what about a live portfolio? Should we expect the same results? In this section

we use the Morningstar style boxes to identify and analyze the universe of small-cap value managers. That

is, we look at a composite of all small-cap value managers as identified by Morningstar.39

The factor exposures shown here are directionally similar to those shown for the hypothetical portfolio we

analyzed in the paper. As expected, we see positive and significant exposure to the market, value and size.40

But an interesting result comes from a comparison of alpha, where we see that alpha goes from zero to

negative in the final model. While this result is different than the stylized example we examined in the

paper, it is consistent with our section on implementable factors. Ultimately, live portfolios face fees,

transaction costs and taxes — all of which fall out of alpha.

Exhibit C1: Analyzing a Composite of Small-Cap Value Managers



Part B: Hypothetical Portfolio Return Decomposition

Source: AQR analysis based on the Morningstar universe of small-cap value mutual funds. The composite returns are net of management and performance fees. The academic explanatory variables are the contemporaneous monthly Fama-French factors for the market (MKT-RF), value (HML),

momentum (UMD), and size (SMB). The portfolio returned 7.5% in excess of cash on average over the period, the market returned 7.8% excess of cash,

HML returned 3.6%, UMD returned 7.3% and SMB returned 1.6%.

39

This composite was obtained from Morningstar as of June 2015. 40

Note that it is not surprising to see a low negative momentum loading as we are only looking at a value portfolio, rather than a 50/50 value/momentum

portfolio (as we did earlier in the paper).


Model 2 (Add HML)

Model 3 (Add UMD)

Model 4 (Add SMB)

Alpha (ann.) 0.0% -1.3% -1.0% -1.8%

t-statistic 0.0 -1.0 -0.8 -2.1

Market Beta 0.96 1.01 1.01 0.95

t-statistic 40.5 42.2 41.3 57.1

HML Beta 0.23 0.23 0.36


UMD Beta -0.02 -0.04

t-statistic -1.0 -2.3

SMB Beta 0.54

t-statistic 22.4

R2 0.80 0.82 0.82 0.92

7.5% 7.9% 7.9% 7.4%

0.9% 0.8% 1.3%

-0.2% -0.3%

0.8%

0.0%

-1.3%-1.0% -1.8%

-3.0%

0.0%

3.0%

6.0%

9.0%

12.0%

Model 1

(Market Control)

Model 2

(Add HML)

Model 3

(Add UMD)

Model 4

(Add SMB)

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

7.4% 8.2%8.3% 7.7%

1.6% 1.7% 2.4%

0.6% 0.5%1.2%

6.1%

3.8%2.9%

1.8%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(add HML)

Model 3

(add UMD)

Model 4

(add SMB)

Market

HML

UMD

SMB

Alpha

An

nu

al E

xc

es

s R

etu

rns

7.5% 7.9% 7.9% 7.4%

0.9% 0.8% 1.3%

-0.2% -0.3%

0.8%

0.0%

-1.3%

-1.0% -1.8%

-3.0%

-1.5%

0.0%

1.5%

3.0%

4.5%

6.0%

7.5%

9.0%

10.5%

12.0%

13.5%

Model 1

(Market Control)

Model 2

(Add HML)

Model 3

(Add UMD)

Model 4

(Add SMB)

Market

HML

UMD

SMB

Alpha


References

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Investing.” Journal of Portfolio Management, 40th

Anniversary edition.

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Finance, Vol. 68, No. 3, 929-985.

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Biographies

Ronen Israel, Principal

Ronen’s primary focus is on portfolio management and research. He was instrumental in helping

to build AQR’s Global Stock Selection group and its initial algorithmic trading capabilities, and he

now also runs the Global Alternative Premia group, which employs various investing styles across

asset classes. He has published in The Journal of Portfolio Management, The Journal of Financial

Economics and elsewhere, and sits on the executive board of the University of Pennsylvania’s

Jerome Fisher Program in Management and Technology. He is an adjunct professor of finance at

New York University, has been a guest speaker at Harvard University, the University of

Pennsylvania, Columbia University and the University of Chicago, and is a frequent conference

speaker. Prior to AQR, Ronen was a senior analyst at Quantitative Financial Strategies Inc. He

earned a B.S. in economics from the Wharton School at the University of Pennsylvania, a B.A.S.

in biomedical science from the University of Pennsylvania’s School of Engineering and Applied

Science, and an M.A. in mathematics, specializing in mathematical finance, from Columbia.

Adrienne Ross, Associate

Adrienne is a member of AQR's Portfolio Solutions Group, where she writes white papers and

conducts investment research. She is also involved in the design of multi-asset portfolios and

engages clients on portfolio construction, risk allocation and capturing alternative sources of

returns. She has published research on how different investments respond to economic

environments in The Journal of Portfolio Management, regional economic factors in The Journal of

Economic Geography and on the Web site of the Federal Reserve Bank of New York. Prior to AQR,

she was a senior associate at PIMCO. She began her career as a researcher at a macroeconomic

think tank in Canada. Adrienne earned a B.A. in economics and mathematics from the University

of Toronto and an M.A. in quantitative finance from Columbia University.


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Measuring Factor Exposures: Uses and Abuses · Uses and Abuses Ronen Israel Principal Adrienne Ross Associate premia). misconstrued as alpha, and investors may be ... classes helps